Consider a one$-$period binomial financial model, with a stock and a bank. The bank offers

the one period interest rate $r0$ for borrowing or depositing. The stock has initial price

$S_0.$ The price of the stock at time $1$ is a random variable $S_1$:$\{H,T\}R.$ This model

has a risk$-$neutral probability measure widetilde$(P),$ with widetilde$(P)(H)=$tilde$(p)$ and widetilde$(P)(T)=$tilde$(q).$

Consider a derivative security $V$ whose payment at time $1$ is a random variable $V_1$ :

$\{H,T\}R.$ Let

$=\frac{V_1(H)-V_1(T)}{S_1(H)-S_1(T)}$

Show that a portfolio $x$ that holds the security $V$ and is short $$ shares of stock will have

a constant value at time $1,$ i$.$e$.x_1(H)=x_1(T).$

This method of reducing the volatility of a portfolio is sometimes referred to as "delta

hedging."